Triangle Ratio Calculator

We've built this triangle ratio calculator so that you can easily find missing angles in triangles using ratios. We'll discuss the math formula for triangle ratio as well as lead you through some examples of solving the angle ratio of a triangle.

If we deal with the angle ratio of a triangle, the formula we need to remember is the one saying that the sum of angles in a triangle is equal to the straight angle, i.e., to 180°:

α + β + γ = 180°

How do you make use of this formula? If you know your angles are in the ratio a:b:c, you can write them as ax, bx, and cx, where x is unknown. And, according to the above formula, we obtain:

ax + bx + cx = 180°

From this, you can determine first the unknown x, and then the angles: ax, bx, and cx. In the next section, we translate these considerations into a step-by-step guide on how to find missing angles in triangles using ratios.

If you know the angles are in the ratio a:b:c and want to determine angles:

1. Write the unknown angles as ax, bx, and cx.

2. Use the fact that they add up to the straight angle: ax + bx + cx = 180°.

3. Simplify the equation: (a + b + c)x = 180°.

4. Compute x = 180°/(a + b + c).

5. Use x to determine the missing angles as ax, bx, and cx.

6. If you need the ratio of sides as well, use the law of sines.

Let's go together through examples to see how it all works in practice:

Example 1

The angle ratio now reads 2:3:4.

Write the angles as 2x, 3x, and 4x, and write the equation for their sum as 2x + 3x + 4x = 180°. After simplification, we obtain 9x = 180° and then x = 20°. As a result, 2x = 40°, 3x = 60°, 4x = 80°. This means our missing angles are 40°, 60°, 80°.

Example 2

Let's say the angles are in the ratio of 1:1:2.

We write the angles as x, x, and 2x. Since their sum is 180°, we get: x+x+2x = 180°. Hence, 4x = 180°, which means that x = 45°. Very easily we get 2x = 90°, and so the missing angles are 45°, 45°, 90°. In particular, this is a right triangle!

Our triangle ratio calculator can work in two ways:

• Enter the angle ratio and get the angles; or
• Enter the angles and get the angle ratio (enter 1 for x).

Make sure to test both possibilities!

As a bonus, we'll also tell you the ratio of side lengths. If you need to convert the side length to angles, go to SSS triangle calculator.

There are two famous angle ratios for triangles: 1:2:3 and 1:1:2. Both these triangles are right triangles and the latter is also isosceles (and we've met it already!). Read more about them here:

• 30 60 90 triangle
• 45 45 90 triangle